Download Use of the SAS Macro Language in Developing Control Chart Limits

USE OF THE SAS MACRO LANGUAGE IN DEVELOPING CONTROL CHART LHIITS
T. P. Walker, IBM General Technology Division, East Fishkill
ABSTRACT
In section II, the steps leading to the
development of the control chart limits are
detailed. Sections III and IV contain an
example as well as an annotated listing of the
SAS Macros, which make up the program.
Control charts are an important tool in applied
statistics, especially in qualjty control.
This report describes the generation of control
chart limits for process data in which the
limits are developed by first grouping the raw
process readings into many relatively small
groups and then, from this grouped data,
obtaining subsets, which are stable with
respect to the standard deviation and the mean.
The statistical theory underlYing the
generation of control chart limits both for
the process mean and for the process variation
is presented in Ref 2, Chapters 3 and 4. See
also Ref 3, Chapters 5, 6, and 7, for an even
more detailed presentation of control chart
theory.
An important feature of this report is the ~
demonstration of the effective use of the SA~
Macro Language in facilitating the calculations
required to develop the stable subsets and the
censor limits. Without the macro language
capabilities, the computations would be long,
tedious, and error-prone.
The calculations presented in the next section
and the example in the last section make use
of a sample of size 9. As will be seen, the
use of a specific sample size makes it
possible to identify exactly the scaling
factors, which are required for generating the
stable sets of data and the control chart
limits. It is hoped that the use of such
specific values will help to clarify the
presentation.
An example is given, and the SAS Macros, which
make up the program, are listed.
I.
INTRODUCTION
Control charts are a well-known tool in quality
control, and the development of control chart
limits is also fairly well known. However,
both the concept of a statistically stable set
of data and the procedure for generating such a
set of data are less well known.
II.
This section describes the generation - from a
set of raw process readings - of a subset of
data, which is in statistical control. There
is a series of 9 steps involved in the
procedure. As will be seen. some of the steps
may be repeated. The sequence proceeds from
generating a set of data, which is stable with
respect to the standard deviation, to
generating a set stable with respect to the
mean, to calculation of control chart limj_ts
for individual values, to generation of the
subset, which is in statistical control.
This paper demonstrates the use of the SAS
Macro Language (Ref. 1, Chapter 15) in
achieving the following objectives:
A)
The generation - from a set of raw data of a subset, which is statistically stable
with respect to the standard deviation.
B)
The generation of another subset, which is
statistically stable with respect to the
mean.
C)
The development of control chart limits
for individual values.
D)
The generation of a subset of the original
raw data readings which is in statistical
control.
CONTROL CHART TECHNIQUES
Ordinarily. a series of repetitive calculations
would be required to achieve objectives A to D.
Programs would have to be rerun several times.
For each run, a user would make a decision
based on the results of the previous run.
However, the SAS Macro Language now makes it
possible to carry out the computations in one
job involving a series of repetitive
calculations but without intervention by the
user.
1)
Assume that N raw process readings are
available. Let variable CT denote an
individual reading. Divide the N
CT-values into A groups, 9 members per
group, i.e., A = N/9 rounded to the
nearest integer.
2)
Compute statistics CTM and CTS for each
of the A groups. where
CTII
CTS
3)
the mean of the set of 9 CTs
the standard deviation of the set
of 9 CTs
Compute MCTM and MCTS, where
MCTM
MCTS
the mean of the set of the A CTMs
the mean of the set of the A CTSs
Put ULSTD
and LLSTD
858
1. 761 X HeTS
0.239 X MCTS
Now construct control chart limits for
individual values and- generate a subset
of data which is in statistical control.
The factors 0.239 and 1.761 are
associated with the sample size of 9 and
are obtained from Table B, in the
Appendix of Ref (2).
4)
5)
8)
From the A groups. delete any group whose
standard deviation value CTS is greater
than ULSTD or less than USTD. As a
result, there will be Al groups left
where Al ~ A. If Al = A. then objective
A has been accomplished, i.e •• we have a
set of stable groups none of whose
standard deviations. CTS. falls outside
the control interval (LLSTD, ULSTD).
On the other hand. suppose that Al < A.
Then. using the Al groups, repeat steps 3
and 4. Compute new values for MCTM.
MeTS. LLSTD, and ULSTD. Next. delete
from the set of Al groups those whose CTS
value falls outside the new interval
(LLSTD, ULSTD). As a result. there
will be A2 groups left, where A2 ~ AI.
If A2 = AI. then we are finished.
Otherwise. repeat steps 3 and 4 using the
A2 groups. At some point, there will be
no more deletions. i.e •• A(K+l) = AK for
some K.
The control chart limits. LLCENSR, and
ULCENSR implement objective C.
9)
MCTS
Put ULMEAN
LLMEAN
III. EXAMPLE (USING THE NOTATION FROM SECTION
Ql
3452
384
the mean of the set of the AI<
eTM's
the mean of the set of the AI<
CTS's
230
13
204
MCTM + 1.094 X MCTS
MCTM
1.094 X MCTS
5
8656
The factor 1.094 is obtained from Table B
in the Appendix of Ref (2), based on the
sample size of 9.
7)
From the set of N raw-process readings.
delete any value CT, which falls outside
the interval (LLCENSR, ULCENSR). The Nl
values. which remain. represent a
process. which is in control. These N1
values implement objective D.
The next section presents an example of
the 9-step procedure just described. The
last section lists the SAS code, which
implements the 9 step procedure.
At this point, a set of AK groups stable
with respect to the standard deviation
has been obtained. Next. we develop a
set stable with respect to the mean.
Let MCTM
the mean of the set of the BT
CTM's.
MCTS
the mean of the set of the BT
CTS's.
Put ULCENSR
MCTM + (3/0.9139) x
MCTS
LLCENSR
MCTM
(3/0.9139) x
MCTS
As with the other factors. 0.9139 is
associated with the sample size of 9 and
is obtained from Table B in the Appendix
of Ref. (2)
(Note: If all groups are deleted. then
the process is totally out of
control. We exclude this possibility)
6)
Let MCTM
974
9265
604
From the AK group. delete any groups whose
mean value CTM falls outside the internal
(LLMEAN, ULMEAN). There will be B1 groups
left where B1 $ AK. If B1 = AK. then
objective B has been accomplished.
Otherwise (if B1 < AK), then using the B1
groups, repeat steps 6 and 7.
11248
7282
2915
As before, at some point there will be no
more deletions. i.e., B(T + 1) = BT for
some T. The BT groups are stable with
respect to the mean. By construction,
they are a subset of the AK groups.
which were stable with respect to the
standard deviation. Hence, the BT groups
are also stable with respect to the
standard deviation. Thus, objectives A
and B have been accomplished.
8656
1421
9167
672
859
N = Number of process readings
(3452/9)
A = Initial number of
groups
AK = Set of groups stable with
respect to the standard deviation
K = Number of iterations required
BT = Set of groups stable with
respect to the mean
T = Number of iterations required
MCTM value for the initial set of
384 groups
MCTS value for the 384 groups
MCTM value for stabilized set of
204 groups
MCTS value for stabilized set of
204 groups
m,CENSR
Upper control chart limit
for individual values
LLCENSR
Lower control chart
limit for individual
values
N1
Subset of the original set
these values represent a
process in statistical
control
Mean value of CT for the original
set of 3452 readings
Standard deviation of CT for the
original set
Mean value of CT for the subset of
2915
Standard deviation of CT for this
subset of 2915.
~'
IV.
IF LLCENSR < CT < ULCENSR;
DATA KEEPER.WOOlN,
SET 14;
~"T4 IS THE CENSORED, UNGROUPED DATA.
SAVE IT IN A SAS DATASET;
PROC MEANS DUA=H;
VAR CT,
TITlEI STATISTICS FOR PROCESS COUHT DATA'
lIlLE2 USIHG UNGROUPED, CENSORED DATA;
HOC CHART DUA"BASEI,
HBAR CT;
TITlE! eAR CHART OF PROCESS COUNTS FOR ORIGINAL DATA -UNGROUP ED:
TITlE~ AND UNCENSORED;
PROC CHART DATA=14;
HBAR CT;
TInH BAR CHART OF PROCESS COUHTS FOR ORIGINAl DATA -UNGROUPED;
TITlEZ AND CENSORED;
PRD~ COHTENTS DATA=~EErER._All_ ~ODS DIRECTORY MAP;
%MEND CENSOR;
SAS CODE FOR PROGRAM
The program consists of a main calling routine
and 4 SAS Macros: CENSOR, STABLEMN, REPEATER.
AND INPUTR. The code is listed at the end of
this section.
Briefly. the programming sequence is as
follows:
1.
2.
3.
4.
Main routine calls INPUTR;
INPUTR calls REPEATER;
REPEATER calls STABLEMN;
Main routine calls CENSOR.
%MACRO STAHlEMN(ARF, TAPAl, TAPB~, 1 APB4.
~DO %WHllE (lARF < 01;
%LH ~ " ~EVAl(&N+U,
PROC MEANS DATA=T9
NOP~INT;
VAR CTM CTS;
OUTPUT OUT=U~
MEAN=MCTM MCTS;
The main calling routine (lines 20500 to 21400)
gets the input data, performs.8 few operations,
and cal~s INPUTR.
ULMEAN=MCTM+(l TAPAl~M"CTSl,
ULSTD =IUPB4"MCTS:
LLMEAN =MCTM- (IT AP A I"MCTS l ,
lLSTD "'"APB~*MCTS:
OAH U4;
IF _H_ = 1
THEN SET U3,
END=EHDTW;
iF IL'"l[AH < CTM < ULM"EAN;
DATA KEEPER.5AVX36;
SET ur.,
%*AFTER THE ITERATION IS COMPLETE, SAVX36 WIll CONTAIN
THE SET OF STA8LE PROCESS DATA -GROUPED.
THIS SET IS STAHE
WITH RESPECT TO THE MEAN:
KEEP FOBS CTM CTS;
PROC MEANS DATA=n
NOPRINT;
VAR elM CTS;
OUTPUT OUT=XTI
~"IITlCTM llTICTS;
PROC MEANS DATA"U~
~OPRIHT;
VAR elM CTS;
OUTPUT OUT=XT~
N=HHCTM "HCTS;
DATA TBOTH2;
MERGE XTl XT4,
IF NllCTM ~= NHCTM
The sample size of 9 and its associated scaling
factors are arguments for INPUTR. These
arguments are passed along to the other macros,
as required. Macro INPUTR groups the data into
sets of 9 (step 1 of II) and calls REPEATER.
Macro REPEATER implements steps 2 to 5 of II.
Notice that REPEATER will rerun until a stable
set of data for the standard deviation is
obtained or until the number of tries exceeds a
user-defined bound (in this case, 20). When
REPEATER finishes, it calls STABLEMN t which
implements steps 6 and 7 of II. Like REPEATER,
STABLEMN repeats until a stable set of data for
the mean is obtained, or until the number of
tries is exceeded.
DO;
TMEAN = 'NO';
H 'N > UN THEH
DO;
TARH = ~EVAUIN);
CALL SYMPUH 'ARF' ,31:
END;
ElSE,
END;
ELSE DO,
When STABLEMN is finished. control returns to
the main program. which calls CENSOR. Macro
CENSOR implements steps 8 and 9 of II.
Additionally, a few other statistics and graphs
are generated. Notice that the arguments for
CENSOR are related to the sample size of 9.
TMEAN
SET T22,
ULCENSR : MCTM + (lCAPll"MGTS):
LlCENSR e MCTM - (&CAPIl"MCTSl;
DATA H;
IF
M - 1 THEM SET H;
SEl BASEl;
=
'O~';
00004700
oua04800
00005100
00005~OO
00~U5~UO
oa~n5600
00005HO
~ 00 G6 a~ 0
oon06100
0000620~
OG~066~0
00006700
00006&00
00001100
OQQ07200
00007300
000076no
00007700
ooooaOOG
ooon~lOo
00008400
00008500
00008700
a000B800
000091.00
ooo~noo
DG009400
B000950B
000097DO
00009800
OOOlooon
OOOlOlon
00010400
00010500
= ~EVAl(lHl'
CALL SYMPUH'ARF',l);
E~D;
SET KEEPER.SA'iX56;
tENO;
PRDC PRINl DATA~TBOTH2;
TlTLEl SIZE OF STABLE DATA SET FOR T~E MEAN,
TITlE2 TKlS SOl IS A SU8SOl OF THE STABLE SET FOR SIGMA;
PROC PRINT DUA=KEEPER.SAVX36;
TITlEI STATISTICS FOR PROCESS COUNT DATA USING A REDUCED;
111LE2 SOl OF DHA GROUPED BY IGROUPX WHICH REPRESEN1S A STABLE;
TITln PROCESS WITH RESPECT TO I~E MEAN;
TITlE~ lHIS SET IS A SUBSET OF THE STABLE SET FOR SIGMA,
XMEND STABLEMN,
XMACRO REPEATER(ARG.ZArAI ,ZAPB,ZAPB~ ,K, lM, LSl,
XDO ~WHIlE (lARG < 6),
Y.lEf K " >:EVAl(&K+ll;
PROC MeANS DATA=Tl
NOPRlHT;
VAR CTM CTS;
ounUT OUT=12
MEAN=MCTM MCTS:
DATA n ;
SET T2;
ULMEAH=MCTMH 12APA1_MCTS) ;
ULSTD =&ZAPM_MCTS,
llM EAN =MCTM- (!UP AI "MCTS 1 ;
LlSTO "&2APB3"I"lCTS;
A listing of the program code follows.
nHIS IS A PROGRAM TO DEVELOP CONTROL CHART LIMITS:
"THIS PROGRAM USE5 THE MACRO LANGUAGE FACHITIES OF SAS AND IS A
VARIATION OF AN OLDER VERSION WHICH USED ONLY MACROS.
THE MACRO LANGUAGE AlLOi<lS THE USE OF EXTENDED AND
REPETITIVE DO lOOPS;
~GlOBAL
GROUPX CAPAI CA?B4 ZAP~l 7.~r~~ 2APB3 CAPll
.I1N
LSG;
,::,ACRO CEN50R(OROUPX,CAPIll,
DATA
BASEl:
SET KoEPER.COMPH;
VCOM~6Z IS THE OnGIHAl DATA -UHGROUHD AND U~CEHSDRED;
PROC MEANS DATAOBA5EI;
VAR CT;
TITLEI STATISTICS FOR FROCESS COUHT DATA;
TITlE2 USIHG ORIGINAl DATA--UNGROUPEO,
DATA Tl;
SET KHPER.FlTEXZ;
;;"FITEXZ IS THE ORIGINAL DATA -GROUPED;
PROC MEANS DATAoTl;
VAR CTM CTS;
OUTPUT OUTen
MEAN=MCTM MCTS;
TITlEl STATISTICS FOR PROCESS COUHT DATA:
THlE2 USrNG ORIGIHAl DATA--GROUPED;
DA1A Tll;
SET KEEPER.SAVX16;
%"SAVX36 IS THE GROUPED PROCESS DATA -STAHE WITH ~ESPECT
TO THE MEAH AND SIGMA;
PROC MEAHS DATA"Tll;
VAR CTM CIS;
OUTPUT OU1=T22
MEAN"MCTM MCIS,
nTlEI STATlSnCS FOR PROCESS COUNT DUA USING A REDUCED;
TlTLE2 SET OF DATA GROUPED BY &GROUPX WHICH REPRESENTS,
~!~~E~3~ STAB~E PROCESS WITH RESPECT TO THE MEAN AND SIGMA:
~,lM);
0000440G
00004500
00000100
00000200
00n00300
0~00040a
n0000500
00a00600
00000100
oooonsoo
00000900
U600lUOO
OOOOllOO
OO~0120G
unOOBOO
OOOOHOO
nonOl50n
00001600
00001706
OUOOIBOO
00nOl90U
IF LLSTD < CTS < UlSTD;
DATA HEPER.CPX6;
SET H;
'I_AFTER THE ITERATION IS COMPlETE, CPX6 WILL CONTAIN
THE SEl OF STABLE PROCESS DATA--GROUPED;
KEEP FO&S eTM CTS;
NOPRIHT,
PROC MEA~S DAH"Tl
VAR elM CIS:
OUTPUT OU1=ZTl
N=NTlCTM HHCTS;
PRO~ MeANS DATA=T4
HOPRINT;
VA~ C1M CTS:
OUIPUT OUT"2H
H=NHCTM NT4CTS,
DATA TBOTH;
~G002000
00002100
ooa0220a
000023UO
00062400
OO~02500
uoua2600
000027~0
00002~0~
~0002~00
O~OOl~~O
00U03100
OOO~3200
U0001300
000U3~00
00003500
00n03600
00001700
OOOO~~OO
00G03900
aD00400a
00G041GO
00004200
860
Oooloson
00011000
00011100
oooll~no
~0011400
00011600
OOO1l700
0001.2000
00012100
00012200
OOU12:l00
00012500
00012600
00012800
00012900
oaon~oo
00013100
00013300
0001HOO
ooon500
00013700
aoonaoo
OGOn9~0
~OGHOO~
000,4100
~ ~ G14 2 0 0
OOOH30n
0001440G
00014500
ooa 1~600
QOG14MO
00014900
00015000
IF ~TICT5 -~ ~HCTS
THE:N
DD;
TSIOMA = 'NO';
IF !K > HS
THE:N
DO;
l~RK
= >;EVALOK),
CAlL SYMPUT('ARG',31;
END;
HSE;
ElSE DOl
r;IOMA
'OK';
~ XEVAl(Ul;
CALL SYMPUT('ARG' 3);
END,
DAlA
n,
SET KEEPER.CPH;
PROC PRI~T DATA=TBOTK'
TITtEl SIZE OF STABLE DATA
FO~ SIGMA;
PRoe PRINT DATA=KEE'P£R.CPX&'
TITlE1 STATISTICS FOR PROCESS COUNT DATA USING A REDWCEP,
TlTLE2 SF.J OF DATA GROWPED BY &GRGwrX WHICH R<PR£SEHT5 A STABLE,
TITL!'; PROCESS WITH RESPECT TO SIGMA;
7.~CONVERGENCE FOR 5InMA-CT$--~AS BEEN OBTAINEO.
NOW USING THIS
STABLE SET OEVELOP A STABlE SET FOR THE MEAN--CTi'\,
sn
SET KoePER.epx"
~STABLEMN(-I, &1.AP~l. UAPS;,HAPB~, D,aM)
%MEHD R"PEAlE~;
%MACRO INPUTR( GROUPX ,CAPAI ,CAPB3 ,CAPB4 .lMN, LOG),
DATA Hsn,
SET B~SEF;
KOBS+],
FDBS=CEIl (K06S/~EVAl (IGROUPX1);
GATA KEEPER.COMP6Z;
SET ~ASE1'
\HG~.?&Z CONTAIHS THE ORIGIHAl, UHGROUPED DATA,
PROC SORT DATA=BASE1;
G001520D
OD0153DD
aDUl5~UD
GD0155DD
ODOl500D
UDa1570~
DDOl58DO
DOOl59DO
DOUl6UDO
DOD L61 D~
D6Dl62UO
DUUH3DO
DODH4~~
ODD1650D
OOU1660D
U001670D
OD0168DD
D D~169DO
DijUllUDO
DOUI710U
DGDI72DD
DODl7HQ
D~ D1 7~ 0 U
ODD17SDD
OD01lODD
UDOl770~
UDUI7800
DnDl7900
DOO
13~DO
OU01810D
ODD1820D
OD01830U
QDD1MOU
OUD1850D
ODD18600
OD01870U
ODnBSOD
UOOlB9DD
UD019CDD
DOH91DO
DOHnou
PRDC MEAHS DATA=BASEi
KOPRINT;
BY fOES;
VAR el;
OUTPUT OUT=OUTER
M[AN=CTM 5TD"CTS;
DATA KEEPER.FlTEXZ;
SET OUTER;
\>FITEX~ CONTAINS TilE OR1GINAl DATA--GROUPED;
DhTA Ti;
SoT OUTER,
7.REPEATER (-1. &CAPAI, &CAPB3,&GAPH4, D ,3lM~, HSG)
Y.M~ND IHPUTR,
DATA
BASEF (KEEP=CT
DATE
TYPE);
SET A5H1.PlOrBOTH
ASH1.PlOTTE2;
TYPE = MGD{MOD(MEFAM.laDl,!O);
GT=lHH FIRSTPAS*IOODO);
PROC SORT DATA~BASEF;
BY TYPE DATE;
PROG PRINT DATA=B~SEF
(O~S=~O);
TITLE! F1RST ~O OBSERVATIOHS--IHPUT DATA:
\lHPUl ~(9. I. U94, D .2'~ 1.761. 2D ,2D)
~C!:"SOR( 9,3.283)
OD0193DD
DD01940D
DDOH5UD
DD0196GD
000197DD
aDnln~D
DDOH~UD
ODe200UD
U002DHD
DD02D20D
DDU2D3ua
00020400
D D~2D5D D
D~02D~U D
DUU2U1DU
DODZ08DO
~00209UO
UUU21DUO
00 D211Q~
ODD2120U
OUD2HOD
OUD214UD
ODD2150Q
REFERENCES
1.
SAS Users Guide, Basics, 1982 edition, SAS
Institute Inc., Box 8000, Cary, NC 27511
2.
R. C. Vaughn, Quality Control, Iowa State
University Press, Ames, Iowa. 1974
3.
I. loJ. Burr. Engineering Statistics and
Quality Control, HcGraw-Hill, New York.
1953
TRADEMARK NOTICE
....
.~
SAS is a registered trademark of the SAS
Institute Inc., Cary, NC. USA.
For More information, contact:
Thomas p. Walker Jr.
IBM Corporation. East Fishkill Facility
Dept. 165. Zip 63A. Route 52
Hopewell Junction, New York 12533
914-894-5293
861